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24 votes

Value of Call Option as Volatility goes to Infinity

The value of a call option does not go to infinity as the volatility goes to infinity. It tends to the discounted value of the forward $F=S_0 e^{(r-q)T}$, which when the dividend yield is zero, ...
Dom's user avatar
  • 2,167
23 votes
Accepted

Gamma Pnl vs Vega Pnl

For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\...
Gordon's user avatar
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21 votes
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Long Gamma vs Vega

Long gamma is being long realized volatility. Long vega is being long implied volatility. Long gamma positions benefit when realized volatility goes up or the actual underlying has volatility. Long ...
AlRacoon's user avatar
  • 6,632
20 votes
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Arbitrage Free Volatility Smile

I generally agree with @dm63's answer: A convex (concave) smile around the forward usually indicates and leptokurtic (platykurtic) implied risk-neutral probability density. Both situations can or ...
LocalVolatility's user avatar
16 votes
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Local vol, stochastic vol, implied vol

Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to ...
Quantuple's user avatar
  • 14.7k
16 votes
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SSR definition in Bergomi in relation to sticky strike and sticky delta

Some Notations It's easy to get lost so let's introduce some notations and let $$ \sigma : (t, S, K, \tau) \to \sigma(K,\tau; S, t) $$ denote the implied volatility smile prevailing at time $t$ ...
Quantuple's user avatar
  • 14.7k
16 votes
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What is the connection between the risk neutral implied density and the real world density?

I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and ...
Kevin's user avatar
  • 16k
14 votes

Forward implied volatility

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward ...
Quantuple's user avatar
  • 14.7k
14 votes
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Option Price vs. Implied Volatility

I think it's interesting to look at this problem graphically also. I get a different answer, depending on whether the option is ITM, ATM, or OTM. In the plot below, all options have 1-year expiry, ...
Jan Stuller's user avatar
  • 6,178
14 votes
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Bergomi: Skew arbitrage

Great question. Let me try to provide some insights and thoughts regarding the points and questions you raised. It may not be a full answer but hopefully it will help connecting the contents in the ...
SI7's user avatar
  • 843
13 votes
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How to exploit calendar arbitrage?

The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and ...
Gordon's user avatar
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13 votes
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Why is realized volatility typically lower than implied volatility?

Consider what happens when IV is lower than realised vol. The person long the IV would make money. So there would ideally be no one selling IV if it's lower than realised vol on an average. Next if ...
nimbus3000's user avatar
13 votes
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Options Market Making Used Implied Volatility Surface

Your question is twofold How a market maker should adjust its quotes on a vol surface with respect to his inventory? How to adjust the vol surface when a new trade is observed on the markets? Let me ...
lehalle's user avatar
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11 votes
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For pricing, what types of Exotic Options are suitable using Local Volatility Model or a Stochastic Volatility Model?

Whenever you use any model to price anything, all you need to do is make sure you model the underlying dynamics that the product you're pricing actually depends on. Any product will be dependent on ...
will's user avatar
  • 2,581
11 votes

Long Gamma vs Vega

Vega (denoted by $\nu$ in what follows) is the first order sensitivity of the option price with respect to volatility $\sigma$. Gamma (denoted by $\Gamma$ in what follows), is the second order ...
Quantuple's user avatar
  • 14.7k
10 votes

What is a regime switch?

Regime switching is another way to describe structural changes in a data series. For example, an inflation timeseries may change states from ARMA to linear as the economy moves from a period of ...
rrg's user avatar
  • 969
10 votes

Why is there greater demand for OTM and ITM options than for ATM options?

Either you or some reference you are following is in error here. At-the-money (or at least near-the-money) options are the most liquidly traded. And trading is much more heavy in out-of-the-money ...
q.t.f.'s user avatar
  • 1,885
10 votes

Implied Volatility of stock on Think or Swim

What they gave you is Newton's formula. If you have a function $f(x)$ then you can find the value $x_0$ such that $f(x_0) = 0$ by this method. It uses the derivative $f'$ which in your case is the ...
Richi Wa's user avatar
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10 votes
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Different versions of sticky strike, moneyness and delta

I feel like your notations are not accurate enough to write what you would like to write. Let $\Sigma(S;K,T)$ denote the implied volatility of a European vanilla of strike $K$ and maturity $T$ now ...
Quantuple's user avatar
  • 14.7k
10 votes
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Different volatility surface ( Local vol, Stochastic vol etc.)

I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric ...
Kevin's user avatar
  • 16k
10 votes
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Realized Variance (realized volatility)

The TLDR; to your question: How can one use realized volatility as a volatility model to do out-of-sample prediction? You extend known models to incorporate additional information procured from high-...
Pleb's user avatar
  • 4,411
9 votes

What does the VIX formula measure and how does it work?

The other answers have given a good qualitative description of what the VIX measures. In this answer, I will try to give a comprehensive quantitative overview of how the VIX formula works. What is the ...
Maximilian Janisch's user avatar
9 votes
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What is implied volatility?

In practice, an implied volatility always refers to the volatility that you need to plug into the Black-Scholes', or Black's, pricing formula to obtain the market price. You may have a different model ...
Gordon's user avatar
  • 21.2k
9 votes
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What is a regime switch?

The idea of regime switching in volatility is rooted in the observation that volatility is usually fairly consistent and "mild", and occasionally very high, say during a market crash. The concept goes ...
pyrex's user avatar
  • 288
9 votes

Why is realized volatility typically lower than implied volatility?

Another theory is that stock indexes do not follow a the lognormal distribution assumption of Black-Scholes. For example U.S. equities measured by the Russell 3000 or S&P 500 have a negative skew ...
Alejandro Sotolongo's user avatar
9 votes

Implied Vol Smile: from Calls, Puts or Both?

In practice, things are actually quite different and a bit more subtle. You really need to differentiate between the underlying being an index or e.g. a single stock. I will try to provide some ...
SI7's user avatar
  • 843
9 votes
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Forward skew generated by Local Vol model

We can demonstrate this via a pricing experiment using QuantLib-Python. I've defined several utility functions in the code block at the bottom of the answer that you will need to replicate the work. ...
StackG's user avatar
  • 3,036
9 votes

Is it possible to have only one volatility surface for american options (that fits both calls and puts)?

Usually, there is only one vol surface (I have never seen or heard of anyone using two). Almost certainly the most advanced commercially available vol surfaces are built by voladynamics. They also ...
AKdemy's user avatar
  • 9,014
8 votes

Is there a good closed-form approximation for Black-Scholes implied volatility?

Let's Be Rational uses exactly two iterations to give full machine accuracy for all inputs. It can be viewed as a three-stage analytical formula if you like. The code is free to download at www....
Peter Jaeckel's user avatar

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